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A sphere of mass m is moving with a velocity 4i-j when it hits a smooth wall and rebounds with velocity i+3j. Find the coefficient of restitution between the sphere and the wall

A sphere of mass m is moving with a velocity 4i-j when it hits a smooth wall and rebounds with velocity i+3j. Find the coefficient of restitution between the sphere and the wall

Grade:12th pass

1 Answers

Sandeep Pathak
askIITians Faculty 25 Points
7 years ago
On a collision between a wall and a sphere, the component of velocity parallel to the wall remains unchanged. In other words, if\vec{u},\vec{v}are the velocities before and after the collision, then the difference between them is always along the normal to the wall as the component along the wall cancels each other. Let\hat{n}=\alpha\hat{i}+\beta\hat{j}}, (\alpha^2+\beta^2 = 1)
be the normal unit vector. Then
\vec{v}-\vec{u} = \left((\vec{v}-\vec{u})\cdot\hat{n} \right )\hat{n}
This equation along with\alpha^2+\beta^2 = 1, gives
\alpha=-\frac{3}{5}, \beta = \frac{4}{5}.
So, we get
\hat{n}=-\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}}
Now,
coefficient of restitution,
e=-\frac{\vec{v}\cdot\hat{n}}{\vec{u}\cdot\hat{n}}=-\frac{-3+12}{-12-4}=\frac{9}{16}

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